Deeply digging the interaction effect in multiple linear regressions using a fractional-power interaction term.

Area of Science:

• Statistics
• Ecology
• Bioinformatics

Background:

• Standard multiple regression models interactions as simple products of variables.
• This linear assumption may not capture complex biological or ecological relationships.

Purpose of the Study:

• Introduce fractional-power interaction regression (FPIR) to model nonlinear interactions.
• Provide a flexible and interpretable alternative to conventional regression techniques.
• Develop an R package for estimating FPIR parameters.

Main Methods:

• Developed FPIR with interaction term \( \beta X_1^M X_2^N \), allowing nonlinear relationships.
• Explored parameter ranges for M and N from -56 to 56.
• Applied FPIR to crested ibis nest site data and 4692 other regression models.

Main Results:

• FPIR models demonstrated significantly lower AIC values compared to regular regressions (\(-302 \pm 5003.5\) vs \(-168.4 \pm 4561.6\)).
• FPIR showed comparable or superior performance against complex models like polynomial regression, GAM, and random forest.
• The effect size of AIC improvement was 0.07 (95% CI: 0.04-0.10).

Conclusions:

• FPIR provides a more flexible and interpretable approach to modeling interactions.
• It effectively maximizes explained variance using fewer degrees of freedom.
• The interactionFPIR R package facilitates the application of FPIR in ecological and statistical analyses.